Next Article in Journal
Applications in Integral Equations through Common Results in C*-Algebra-Valued Sb-Metric Spaces
Next Article in Special Issue
Applications Laguerre Polynomials for Families of Bi-Univalent Functions Defined with (p,q)-Wanas Operator
Previous Article in Journal
Spanning k-Ended Tree in 2-Connected Graph
Previous Article in Special Issue
New Criteria for Convex-Exponent Product of Log-Harmonic Functions
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

A q-Analog of the Class of Completely Convex Functions and Lidstone Series

by
Maryam Al-Towailb
1,* and
Zeinab S. I. Mansour
2
1
Department of Computer Science and Engineering, College of Applied Studies and Community Service, King Saud University, Riyadh 11451, Saudi Arabia
2
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
*
Author to whom correspondence should be addressed.
Axioms 2023, 12(5), 412; https://doi.org/10.3390/axioms12050412
Submission received: 14 February 2023 / Revised: 8 March 2023 / Accepted: 19 March 2023 / Published: 24 April 2023
(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

Abstract

:
This paper introduces a q-analog of the class of completely convex functions. We prove specific properties, including that q-completely convex functions have convergent q-Lidstone series expansions. We also provide a sufficient and necessary condition for a real function to have an absolutely convergent q-Lidstone series expansion.
MSC:
05A30; 41A58; 39A70; 40A05

1. Introduction

In 1929, Lidstone [1] introduced a generalization of Taylor’s theorem that approximates an entire function f in a neighborhood of two points instead of one. That is
f ( x ) = n = 0 f ( 2 n ) ( 1 ) Λ n ( x ) + f ( 2 n ) ( 0 ) Λ n ( 1 x ) ,
where Λ n ( x ) is a unique polynomial of degree 2 n + 1 , and called a Lidstone polynomial. In [2], Whittaker proved that an entire function of an exponential type of less than π has a convergent Lidstone series expansion in any compact set of the complex plane. Buckholtz and Shaw [3] provided some conditions for (1) to hold. Other authors worked on this problem (see, e.g., [4,5,6,7,8,9,10]). They presented different sufficient and necessary conditions for the representation of functions by this series. We mention, in particular, the result of Widder [10]. He proved that if f is a real-valued function satisfying
( 1 ) k f ( 2 k ) ( x ) 0 ( k N 0 )
in an interval of length greater than π , then it has a Lidstone series expansion (1) (such a function is known as completely convex). Furthermore, he defined the class of minimal completely convex functions, and then he proved that a real-valued function f ( x ) could be expanded in an absolutely convergent Lidstone series if and only if it is the difference of two minimal completely convex functions.
Recently, the Lidstone expansion theorem was generalized in quantum calculus (as can be seen in [11,12,13,14,15,16,17]). The quantum calculus (Jackson calculus or q-calculus [18]) is an extension of the traditional calculus, and it has been used by many researchers in different branches of science and engineering (as can be seen in, e.g., [19,20,21,22,23,24]). It has a lot of applications in different mathematical areas such as orthogonal polynomials, number theory, hypergeometric functions, theory of finite differences, gamma function theory, Sobolev spaces, Bernoulli and Euler polynomials, operator theory, and quantum mechanics. For the basic definitions and notations applicable in the q-calculus, see Section 2.
In [11], Ismail and Mansour proved the following q-analog of the Lidstone expansion theorem.
Theorem 1.
Assume that the function f ( z ) is an entire function of q 1 -exponential growth of order 1 and a finite type α less than ξ 1 , or it is an entire function of q 1 -exponential growth of an order of less than 1. Then, f ( z ) has a convergent q-Lidstone representation
f ( z ) = n = 0 D q 1 2 n f ( 1 ) A n ( z ) D q 1 2 n f ( 0 ) B n ( z ) ,
where ( A n ) n and ( B n ) n are the q-Lidstone polynomials defined, respectively, by the generating functions
E q ( z w ) E q ( z w ) E q ( w ) E q ( w ) = n = 0 A n ( z ) w 2 n ,
E q ( z w ) E q ( w ) E q ( z w ) E q ( w ) E q ( w ) E q ( w ) = n = 0 B n ( z ) w n [ n ] q ! .
Moreover, A 0 ( z ) = z , B 0 ( z ) = 1 z , and for n N , A n ( z ) and B n ( z ) satisfy the q-difference equation
D q 1 2 y n ( z ) = y n 1 ( z ) w i t h y n ( 0 ) = y n ( 1 ) = 0 .
In [16], AL-Towailb and Mansour proved that the condition
D q 1 n f ( 0 ) = o ( ξ 1 n ) as n
is both sufficient and necessary for expanding an entire function f ( z ) in the q-Lidstone series
f ( 1 ) A 0 ( z ) f ( 0 ) B 0 ( z ) + D q 1 2 f ( 1 ) A 1 ( z ) D q 1 2 f ( 0 ) B 1 ( z ) + ,
and we noted that Condition (7) is insufficient for the convergence of the following arrangement of the q-Lidstone series:
n = 0 D q 1 2 n f ( 1 ) A n ( z ) n = 0 D q 1 2 n f ( 0 ) B n ( z ) ,
and not necessary for the convergence of (3). This paper aimed to obtain a sufficient and necessary condition for a real-valued function to have an absolutely convergent q-Lidstone series expansion (3). To achieve this aim, we introduced generalizations for the class of completely convex functions (2) on a closed interval of form [ 0 , a ] ( a > 0 ) , and the class of minimal completely convex functions on the interval [ 0 , 1 ] . This paper is organized as follows. The following section gives the essential notions and basic definitions of q-calculus. Section 3 contains some properties and basic results on q-Lidstone polynomials, which we need in our investigation. In Section 4, we define a q-analog of the class of completely convex functions for the difference operator D q 1 . Then, we study the relation of this class to a problem of the representation of functions by the q-Lidstone series. In Section 5, we provide a necessary and sufficient condition for a real function to have an absolutely convergent q-Lidstone series expansion.

2. Preliminaries

In this section, we recall some definitions, notations, and results in the q-calculus, which we need in our investigations (see [25]).
Throughout this paper, q is a positive number less than one, and we use the following standard notations:
N : = { 1 , 2 , 3 , } , N 0 : = { 0 , 1 , 2 , } = N { 0 } .
The sets A q and A q * are defined by A q : = { q n : n N 0 } and A q * : = A q { 0 } . For a C , n N 0 ,
( a ; q ) = j = 0 ( 1 a q j ) , ( a ; q ) n : = ( a ; q ) ( a q n ; q ) ,
and the q-numbers [ n ] q and q-factorial [ n ] q ! are defined by
[ n ] q = 1 q n 1 q , [ n ] q ! = k = 1 n [ k ] q .
Let μ C . A set A C is called μ -geometric set if μ z A for any z A . If f is a function defined on a q-geometric set A, then Jackson’s q-difference operator is defined by
D q f ( z ) = f ( z ) f ( q z ) ( 1 q ) z , z A { 0 } ; f ( 0 ) , z = 0 ,
provided that f is differentiable at zero. Furthermore, Jackson [26] introduced the following q-integrals for a function f defined on a q-geometric set A:
a b f ( t ) d q t : = 0 b f ( t ) d q t 0 a f ( t ) d q t ( a , b R ) ,
where
0 z f ( t ) d q t : = ( 1 q ) n = 0 z q n f ( z q n ) ,
provided that the series converges at z = a and z = b .
Jackson’s q-trigonometric functions Sin q z and Cos q z are defined by
Sin q z : = n = 0 ( 1 ) n q n ( 2 n + 1 ) ( q ; q ) 2 n + 1 ( z ( 1 q ) ) 2 n + 1 , Cos q z : = n = 0 ( 1 ) n q n ( 2 n 1 ) ( q ; q ) 2 n ( z ( 1 q ) ) 2 n ,
where E q ( · ) is one of Jackson’s q-exponential function defined by
E q ( z ) = n = 0 q n ( n 1 ) 2 ( z ( 1 q ) ) n ( q ; q ) n = ( z ( 1 q ) ; q ) ( z C ) .
We use { ξ k } k N to denote the positive zeros of Sin q z arranged in increasing order of magnitude. One can verify that Sin q z has no zeroes on | z | < q 3 / 2 , i.e., the first positive zeros ξ 1 > q 3 / 2 .
Lemma 1.
For any x [ 0 , 1 ] , we have
Sin q ξ 1 x ξ 1 x .
Proof. 
Let f ( x ) = ξ 1 x Sin q ξ 1 x , x [ 0 , 1 ] . Then, D q 1 f ( x ) = ξ 1 ( 1 Cos q ξ 1 x ) 0 . Therefore, by using (8), we obtain
f ( x ) f ( x q ) ( x [ 0 , 1 ] ) ,
which implies f ( x ) lim n f ( q n x ) = 0 . Then, Inequality (11) holds. □

3. Some Results on q -Lidstone Polynomials

We start this section by recalling some properties of the q-Lidstone polynomials A n ( x ) and B n ( x ) from [14,16,17], for which we need to prove the main results.
Proposition 1.
([16]). Let { ξ k } k N be the sequence of the positive zeros of Sin q ( x ) and m N 0 . Then,
( 1 ) n 1 A n ( x ) = 2 Sin q ( ξ 1 x ) ξ 1 2 n + 1 Sin q ( ξ 1 ) + O ( ξ 2 ( 2 n + 1 ) ) ;
( 1 ) n 1 B n ( x ) = Sin q ( ξ 1 x ) Cos q ( ξ 1 ) ( 1 q ) ( ξ 1 ) 2 n + 1 Sin q ( ξ 1 ) + O ( ξ 1 2 n ( 2 n ) m ) ,
for a sufficiently large n.
Proposition 2.
([17]). If f C q 2 n ( [ 0 , 1 ] ) , then
f ( x ) = m = 0 n 1 D q 1 2 m f ( 1 ) A m ( x ) D q 1 2 m f ( 0 ) B m ( x ) + 0 1 G n ( x , q t ) D q 1 2 n f ( q 2 t ) d q t ,
where
G ( x , t ) = G 1 ( x , t ) = q t ( 1 x ) , 0 t < x 1 ; q x ( 1 t ) , 0 x < t 1 ,
G n ( x , q t ) = 0 1 G ( x , q y ) G n 1 ( q y , q t ) d q y ( n N ) .
Moreover,
0 1 G n ( x , q t ) d q t = A n ( x ) B n ( x ) ( n N ) .
Remark 1.
([14]). For x [ 0 , 1 ] and n N 0 , we have
( 1 ) n A n ( x ) 0 a n d ( 1 ) n 1 B n ( x ) 0 .
Proposition 3.
Let ξ 1 be the smallest positive zero of Sin q ( x ) . Then, there exist some constants M 1 and M 2 and a positive integer n 0 such that the following inequalities hold
0 ( 1 ) n A n ( x ) M 1 ξ 1 2 n ;
0 ( 1 ) n 1 B n ( x ) M 2 ξ 1 2 n ,
for all x [ 0 , 1 ] and n n 0 .
Proof. 
From (12), there is a positive real number C 1 and n 0 N such that
| ( 1 ) n 1 A n ( x ) 2 Sin q ( ξ 1 x ) ξ 1 2 n + 1 Sin q ( ξ 1 ) | C 1 ξ 2 2 n ,
for all x [ 0 , 1 ] and n n 0 . Consequently,
0 ( 1 ) n A n ( x ) C 1 ξ 2 2 n 2 Sin q ( ξ 1 x ) ξ 1 2 n + 1 Sin q ( ξ 1 ) .
Note that ξ 1 < ξ 2 and Sin q ( ξ 1 x ) is bounded on [ 0 , 1 ] . Then, from (22), we obtain
0 ( 1 ) n A n ( x ) C 1 ξ 1 2 n + 2 ξ 1 2 n + 1 | Sin q ( ξ 1 x ) Sin q ( ξ 1 ) | C 1 ξ 1 2 n + C 2 ξ 1 2 n = M 1 ξ 1 2 n .
Similarly, we obtain (20) from (13). □
Proposition 4.
There exists a constant M such that
0 0 1 ( 1 ) n G n ( x , q t ) d q t M ξ 1 2 n .
Proof. 
The proof follows immediately from Equation (17) and Proposition 3. □
Proposition 5.
For any fixed point x 0 ( 0 , 1 ) and sufficiently large n, there exist some constants M 1 and M 2 such that
( 1 ) n A n ( x 0 ) M 1 ξ 1 2 n ;
( 1 ) n 1 B n ( x 0 ) M 2 ξ 1 2 n .
Proof. 
From (12), we obtain
( 1 ) n A n ( x ) ξ 1 2 n + 1 = L ( x ) + O ( ( ξ 1 ξ 2 ) 2 n + 1 ) ( n ) ,
where L ( x ) = 2 Sin q ( ξ 1 x ) Sin q ( ξ 1 ) . Notice, for any fixed x 0 ( 0 , 1 ) , L ( x 0 ) > 0 and
lim n ( 1 ) n A n ( x 0 ) ξ 1 2 n + 1 = L ( x 0 ) .
This implies that the sequence ( 1 ) n A n ( x 0 ) ξ 1 2 n + 1 is bounded below by a positive number. I.e., (24) holds. Similarly, we obtain the Inequality (25) from (13). □
Now, using the previous results, we prove the following theorem.
Theorem 2.
If the series
S = a 0 A 0 ( x ) + b 0 B 0 ( x ) + a 1 A 1 ( x ) + b 1 B 1 ( x ) +
converges for a single value x 0 ( 0 , 1 ) , then the series n = 0 ( 1 ) n a n + b n ξ 1 2 n is absolutely convergent.
Proof. 
Since the series (26) converges for x 0 ( 0 , 1 ) , we have
lim n a n A n ( x 0 ) = 0 , lim n b n B n ( x 0 ) = 0 .
Then, from the inequalities (24) and (25), we obtain
a n = O ( ξ 1 2 n ) and b n = O ( ξ 1 2 n ) .
From (12), (13), and (27), we conclude that the series
S 1 = n = 0 a n A n ( x 0 ) + 2 ( 1 ) n Sin q ( ξ 1 x 0 ) ξ 1 2 n + 1 Sin q ( ξ 1 ) + b n B n ( x 0 ) + ( 1 ) n Cos q ξ 1 Sin q ( ξ 1 x 0 ) ( 1 q ) ξ 1 2 n + 1 Sin q ( ξ 1 )
converges absolutely. This implies that S 1 S is also convergent. Notice that
S 1 S = n = 0 2 Sin q ( ξ 1 x 0 ) ξ 1 Sin q ( ξ 1 ) ( 1 ) n ξ 1 2 n a n + Cos q ξ 1 Sin q ( ξ 1 x 0 ) ( 1 q ) ξ 1 Sin q ( ξ 1 ) ( 1 ) n ξ 1 2 n b n > 2 Sin q ( ξ 1 x 0 ) ξ 1 Sin q ( ξ 1 ) n = 0 ( 1 ) n ξ 1 2 n a n + ( 1 ) n ξ 1 2 n b n .
Therefore, we obtain the result. □

4. A q -Analog of Completely Convex Function

In this section, by C q [ 0 , a ] , we mean the space of all functions defined on [ 0 , a ] such that D q 1 n f ( x ) is defined and continuous at zero.
Definition 1.
A real-valued function f, defined on the interval [ 0 , a ] ( a > 0 ) , is said to be a q-completely convex function if f C q [ 0 , a ] and
( 1 ) n D q 1 2 n f ( a q k ) 0 ( f o r   a l l { n , k } N 0 ) .
Example 1.
The functions f ( x ) = Sin q ξ 1 x , defined in (9), are q-completely convex on the interval [ 0 , 1 ] . Indeed, one can verify that
( 1 ) n D q 1 2 n f ( x ) = ( 1 ) n D q 1 2 n Sin q ξ 1 x = ξ 1 2 n Sin q ( ξ 1 x ) > 0 ,
for all x [ 0 , 1 ] and n N 0 .
In the following, we prove certain properties of q-completely convex functions.
Proposition 6.
If a function f C q [ 0 , a ] is q-completely convex, then
( 1 ) n D q 1 2 n f ( 0 ) 0 ( n N 0 ) .
Proof. 
The proof follows directly by taking the limit as k in (28) and using that D q 1 2 n f is continuous at zero for all n N 0 . □
Proposition 7.
Let f C q ( 0 , 1 ) be a q-completely convex function on [ 0 , 1 ] . Then, for a sufficiently large n, we have
D q 1 2 n f ( 0 ) = O ( ξ 1 2 n ) ;
D q 1 2 n f ( 1 ) = O ( ξ 1 2 n ) .
Proof. 
From Proposition 1 and Inequality (28), every term of (14) is non-negative. Therefore,
0 A n ( x ) D q 1 2 n f ( 0 ) f ( x ) ;
0 ( B n ( x ) ) D q 1 2 n f ( 1 ) f ( x ) ( x [ 0 , 1 ] ; n N 0 ) .
Thus, by using (19) and (33), we obtain
0 ( 1 ) n D q 1 2 n f ( 0 ) f ( x 0 ) ( 1 ) n A n ( x 0 ) K ξ 1 2 n ( n ) ,
for some constant K > 0 and x 0 ( 0 , 1 ) . Then, we have (31). Similarly, we obtain the asymptotic behavior in (32). □
Proposition 8.
Let f be a q-completely convex function on [ 0 , 1 ] . Then, there exists a positive constant C such that for all x A q
0 ( 1 ) n D q 1 2 n f ( x ) C ξ 1 x 2 n ,
where ξ 1 is the smallest positive zero of Sin q ( x ) .
Proof. 
If f is q-completely convex on [ 0 , 1 ] , then it is q-completely convex on [ 0 , x ] for all x A q . Consequently, the function f ˜ ( t ) : = f ( x t ) is q-completely convex on [ 0 , 1 ] . Therefore, from Proposition (7), we have
0 ( 1 ) n D q 1 2 n f ˜ ( 1 ) = ( 1 ) n x 2 n D q 1 2 n f ( x ) = O ( ξ 1 2 n ) ,
which is nothing else but (35). □
Lemma 2.
Let f ( x ) and D q 1 2 f ( x ) be non-negative on A q * , and continuous at 0. Assume that there exists a number x 0 A q such that f ( x 0 ) α ( α R ) . Then,
f ( x ) ( 1 + q ) α ( 1 q ) x 0 , f o r   a l l x A q * .
Proof. 
First, let x A q * and x x 0 . Then, by using the assumption D q 1 2 f ( x ) 0 , we have
x 0 x D q 2 f ( t q 2 ) d q t 0 .
Therefore, D q f ( x ) D q f ( x 0 ) , and
x 0 x D q f ( t ) d q t ( x x 0 ) D q f ( x 0 ) ( x A q * , x 0 x ) .
Since f ( x ) 0 on A q * , from (8) and Inequality (36), we obtain
f ( x ) f ( x 0 ) + ( x x 0 ) ( 1 q ) x 0 f ( x 0 ) = x x 0 q ( 1 q ) x 0 f ( x 0 ) < α ( 1 q ) x 0 ,
for all x A q * and x 0 x . Similarly, if x A q * and x < x 0 , then
f ( x ) x 0 x ( 1 q ) x 0 f ( q x 0 ) < f ( q x 0 ) ( 1 q ) x 0 .
On the other hand, since D q 1 2 f ( x ) 0 , we have
( 1 + q ) f ( q x ) q f ( x ) + f ( q 2 x ) ( x A q * ) .
Therefore, from the condition f ( x ) 0 , we obtain
( 1 + q ) f ( q x ) q f ( x q ) + f ( q x ) > f ( q x ) ( x A q * ) .
So, from the inequalities (38) and (39), we obtain
f ( x ) < ( 1 + q ) α ( 1 q ) x 0 ( x A q * , x < x 0 ) .
Hence, the relations (37) and (40) yield the required result. □
Corollary 1.
If f C q [ 0 , 1 ] is a q-completely convex function, then there exists a positive constant M such that
0 ( 1 ) n D q 1 2 n f ( x ) M ξ 1 2 n ( n N 0 , x A q * ) .
Proof. 
The proof follows from Proposition 8 and Lemma 2 by taking x 0 = 1 and M = 1 + q 1 q C . □
Lemma 3.
If f C q [ 0 , 1 ] is a q-completely convex function on [ 0 , 1 ] , then there exists a constant K > 0 such that
| D q 1 n f ( x ) | K ξ 1 n ( x A q * ) ,
where ξ 1 is the smallest positive zero of Sin q ( z ) .
Proof. 
From Corollary 1, it suffices to prove (42) when n is an odd integer. We set g ( x ) = ( 1 ) n D q 1 2 n f ( x ) . Since f ( x ) is a q-completely convex on 0 x 1 , again from Corollary 1, there exists the constant M > 0 (independent of n) such that for all x A q *
0 g ( x ) M ξ 1 2 n ,
0 D q 1 2 g ( x ) M ξ 1 2 n + 2 .
Therefore, for every x A q * { 1 } , we have
0 q x q 2 D q 1 2 g ( t ) d q t M q ( q x ) ξ 1 2 n + 2 .
So, by using the fundamental theorem of the q-calculus, we obtain
0 ( 1 ) n D q 1 2 n + 1 f ( x ) ( 1 ) n D q 1 2 n + 1 f ( 1 ) M ξ 1 2 n + 2 ,
and hence,
( 1 ) n D q 1 2 n + 1 f ( 1 ) ( 1 ) n D q 1 2 n + 1 f ( x ) ( 1 ) n D q 1 2 n + 1 f ( 1 ) + M ξ 1 2 n + 2 ,
for all x A q * { 1 } . Consequently,
| D q 1 2 n + 1 f ( x ) | | D q 1 2 n + 1 f ( 1 ) | + M ξ 1 2 n + 2 .
On the other hand, since D q 1 2 g ( x ) < 0 , one can verify that for all x A q *
( 1 + q ) g ( x q ) g ( x ) + q g ( x q 2 ) ,
and then
q g ( x q ) ( 1 + q ) g ( x ) g ( q x ) ( x A q * ) .
Thus, if x = 1 , we obtain
( D q 1 2 n f ) ( 1 q ) = ( 1 ) n ( D q 1 2 n f ) ( 1 q ) ( 1 + q ) q D q 1 2 n f ( 1 ) .
Hence, from (8), (43) and (45), we have
D q 1 2 n + 1 f ( 1 ) = | D q 1 g ( 1 ) | | g ( 1 ) | + | g ( 1 / q ) | 1 / q 1 2 q + 1 q M ξ 1 2 n .
However, ξ 1 > q 3 / 2 , this implies
D q 1 2 n + 1 f ( 1 ) q ( 2 q + 1 ) M ξ 1 2 n + 1 .
By substituting (47) in (44), we obtain
| D q 1 2 n + 1 f ( x ) | q ( 2 q + 1 ) M ξ 1 2 n + 1 + M ξ 1 2 n + 2 M 1 M ξ 1 2 n + 1 ,
for all n N and x A q * , where M 1 = q ( 2 q + 1 ) + q 3 / 2 .
Since D q 1 2 n + 1 f ( x ) is continuous at zero, then we obtain D q 1 2 n + 1 f ( x ) = O ( ξ 1 2 n + 1 ) for a sufficiently large n. This completes the proof. □
Theorem 3.
Let f C q [ 0 , 1 ] be a q-completely convex on [ 0 , 1 ] . If f is analytic at zero, then the following q-Lidstone series expansion holds for all x [ 0 , 1 ] .
f ( x ) = n = 0 D q 1 2 n f ( 1 ) A n ( x ) D q 1 2 n f ( 0 ) B n ( x ) .
Moreover, f ( x ) is the restriction of an entire function of q 1 -exponential growth of order 1 and a finite type less than ξ 1 and the expansion (48) holds for all x on the entire complex plane.
Proof. 
Since f is analytic at 0, there exists 0 < c < 1 and the open interval Ω c = ( c , c ) such that f ( x ) has the Maclaurin series expansion
f ( x ) = n = 0 f ( n ) ( 0 ) n ! x n = n = 0 q n ( n 1 ) 2 D q 1 n f ( 0 ) [ n ] q ! x n ( x Ω c ) .
From Lemma 3, there exists a constant K such that
| f ( x ) | n = 0 | q n ( n 1 ) 2 D q 1 n f ( 0 ) [ n ] q ! x n | K n = 0 q n ( n 1 ) 2 ( ξ 1 x ) n [ n ] q ! = K E q ( ξ 1 x ) ,
where E q ( . ) is Jackson’s q-exponential function defined in (10). Notice that, by the known properties of E q ( . ) (see [11]), E q ( x ) is an entire function that has a q 1 -exponential growth of order 1, and it converges everywhere in the complex plane. Therefore, f ( x ) is the restriction of an entire function of q 1 -exponential growth of order 1 and a finite type less than ξ 1 . So, according to Theorem 1, we obtain the result. □

5. A q -Analog of Minimal Completely Convex Function

Definition 2.
A real-valued function f C q [ 0 , 1 ] is a minimal q-completely convex on [ 0 , 1 ] if it is q-completely convex in the interval [ 0 , 1 ] , and if the function g ( x ) = f ( x ) ϵ Sin q ξ 1 x is not q-completely convex for any ϵ > 0 .
For example, the function f ( x ) = Sin q x is a minimal q-completely convex in 0 x 1 while the function f ( x ) = Sin q ξ 1 x is not because for any 0 < ϵ < 1 and x ( 0 , 1 ) ,
( 1 ) n D q 1 2 n Sin q ξ 1 x ϵ Sin q ξ 1 x = ( 1 ϵ ) ξ 1 2 n Sin q ( ξ 1 x ) > 0 .
Theorem 4.
Let n N 0 , ( a n ) n and ( b n ) n be two sequences of non-negative integers. Assume that the series
n = 0 ( 1 ) n a n A n ( x ) ( 1 ) n b n B n ( x )
converges to a function f ( x ) , 0 x 1 . Then, f ( x ) is a minimal q-completely convex on the interval [ 0 , 1 ] .
Proof. 
From the assumption, we have
f ( x ) = n = 0 ( 1 ) n a n A n ( x ) ( 1 ) n b n B n ( x ) , 0 x 1 .
Taking the q 1 -derivative for (50) 2 k times and using (6), we obtain
( 1 ) k D q 1 2 k f ( x ) = n = k ( 1 ) n k a n A n k ( x ) ( 1 ) n k b n B n k ( x ) = m = 0 ( 1 ) m a m + k A m ( x ) ( 1 ) m b m + k B m ( x ) .
From Proposition 5, since ( a n ) n and ( b n ) n are positive sequences, the right-hand side of Equation (52) is non-negative, and f ( x ) is q-completely convex in [ 0 , 1 ] . On the other hand, from Proposition 3 and Equation (52), there exists a constant M > 0 such that
( 1 ) k D q 1 2 k f ( x ) M m = 0 a m + k + b m + k ξ 1 2 m = M ξ 1 2 k n = k a n + b n ξ 1 2 n .
According to Theorem 2, the power series T k = n = k a n + b n ξ 1 2 n converges to zero as k . Hence, for given ϵ > 0 and x 0 A q , there exists an integer k 0 N such that
M T k ϵ Sin q ( ξ 1 x 0 ) < 0 ( k k 0 ) .
This implies from (53) that the function
( 1 ) k D q 1 2 k f ( x ) ϵ Sin q ( ξ 1 x ) = ( 1 ) k D q 1 2 k f ( x ) ϵ ξ 1 2 k Sin q ( ξ 1 x )
is negative at x 0 . Therefore, the function f is a minimal q-completely convex in [ 0 , 1 ] . □
Theorem 5.
If f ( x ) is a minimal q-completely convex function on [ 0 , 1 ] , then it can be expanded into a convergent q-Lidstone series:
f ( x ) = f ( 1 ) A 0 ( x ) f ( 0 ) B 0 ( x ) + D q 1 2 f ( 1 ) A 1 ( x ) D q 1 2 f ( 0 ) B 1 ( x ) + .
Proof. 
We denote by S n ( x ) the nth partial sum of the series (54). Then, from the hypothesis on f ( x ) and Equation (14), we obtain
S n ( x ) f ( x ) ( 0 x 1 , n N 0 ) .
Moreover, for each x, S n ( x ) is a non-decreasing function of n. Thus, lim n S n ( x ) exists and tends towards some function. To prove the result, we prove that
lim n S n ( x ) = f ( x ) ( x [ 0 , 1 ] ) .
Suppose the contrary, and assume that for some x 0 [ 0 , 1 ]
f ( x 0 ) lim n S n ( x 0 ) = > 0 .
Then, by using Equation (14), we have
f ( x 0 ) S 2 n ( x 0 ) = 0 1 G n ( x 0 , q t ) D q 1 2 n f ( q 2 t ) d q t ( n N ) .
Since f ( x ) is a minimal q-completely convex function on [ 0 , 1 ] , then f ( x ) ϵ Sin q ξ 1 x is not q-completely convex in 0 x 1 for any ϵ > 0 . That is, there exists n 0 N and t 0 A q ,
( 1 ) n 0 D q 1 2 n 0 f ( t 0 ) ϵ ξ 1 2 n 0 Sin q ( ξ 1 t 0 ) < 0 .
From Inequality (11), we have
( 1 ) n 0 D q 1 2 n 0 f ( t 0 ) < ϵ ξ 1 2 n 0 + 1 t 0 .
By applying Lemma 2 on the function g ( x ) = ( 1 ) n 0 D q 1 2 n 0 f ( x ) , we obtain
( 1 ) n 0 D q 1 2 n 0 f ( t ) 1 + q 1 q ϵ ξ 1 2 n 0 + 1 ( t A q ) .
Therefore, by choosing ϵ < 1 q ( 1 + q ) ξ 1 M , where M is the constant of Proposition 4, we obtain
0 0 1 G n 0 ( x 0 , q t ) D q 1 2 n 0 f ( q 2 t ) d q t < ,
which contradicts Inequality (55), and then the result is proved. □
The following theorem is the main result of this section.
Theorem 6.
A real function f ( x ) can be represented by an absolutely convergent q-Lidstone series if and only if it is the difference of two minimal q-completely convex functions on [ 0 , 1 ] .
Proof. 
First, assume that f ( x ) = g ( x ) h ( x ) , where g ( x ) and h ( x ) are both minimal q-completely convex functions on [ 0 , 1 ] . According to Theorem 5, we have
g ( x ) = n = 0 D q 1 2 n g ( 1 ) A n ( x ) D q 1 2 n g ( 0 ) B n ( x ) ,
h ( x ) = n = 0 D q 1 2 n h ( 1 ) A n ( x ) D q 1 2 n h ( 0 ) B n ( x ) .
Notice that each series only has positive terms. Thus, by subtracting (57) from (56), we obtain an absolutely convergent q-Lidstone series whose sum is f ( x ) .
Conversely, assume that f ( x ) can be represented by an absolutely convergent q-Lidstone series
f ( x ) = n = 0 D q 1 2 n f ( 1 ) A n ( x ) D q 1 2 n f ( 0 ) B n ( x ) .
Set a n = D q 1 2 n f ( 1 ) , b n = D q 1 2 n f ( 0 ) , and
g ( x ) = n = 0 ( 1 ) n { | a n | ( 1 ) n a n } A n ( x ) + ( 1 ) n + 1 { | b n | ( 1 ) n b n } B n ( x ) ,
h ( x ) = n = 0 ( 1 ) n | a n | A n ( x ) + ( 1 ) n + 1 | b n | B n ( x ) .
Since series in (58) is absolutely convergent, then the two series in (59) and (60) both converge. Furthermore, note that every term of these series is positive. Hence, by using Theorem 4, g ( x ) and h ( x ) are minimal q-completely convex functions on [ 0 , 1 ] . Since f ( x ) = h ( x ) g ( x ) , the proof is complete. □

6. Conclusions

We introduced the class of q-completely convex functions in the interval [ 0 , a ] , with the functions satisfying the inequality
( 1 ) n D q 1 2 n f ( a q k ) 0 ( { n , k } N 0 ) ) .
This class of functions is a generalization of the class of completely convex functions introduced by Widder [10]. First, we presented some properties of a q-completely convex function, and then we proved that such a function could be expanded in a convergent q-Lidstone series:
f ( x ) = n = 0 D q 1 2 n f ( 1 ) A n ( x ) D q 1 2 n f ( 0 ) B n ( x ) .
Furthermore, we obtained a necessary and sufficient condition for a function f ( x ) to have an absolutely convergent q-Lidstone series expansion by introducing the class of minimal q-completely convex functions.

Author Contributions

M.A.-T. and Z.S.I.M. equally and significantly contributed to writing this article. All authors have read and agreed to the published version of the manuscript.

Funding

Research Center of the Female Scientific and Medical Colleges, Deanship of Scientific Research, King Saud University.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

This research project was supported by a grant from the ”Research Center of the Female Scientific and Medical Colleges”, Deanship of Scientific Research, King Saud University. The authors would like to thank the editor and the referees for their helpful comments and suggestions that improved this article.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Lidstone, G. Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types. Proc. Edinb. Math. Soc. 1929, 2, 16–19. [Google Scholar] [CrossRef]
  2. Whittaker, J.M. On Lidstone’ series and two-point expansions of analytic functions. Proc. Lond. Math. Soc. 1934, 2, 451–469. [Google Scholar] [CrossRef]
  3. Buckholtz, J.D.; Shaw, J.K. On functions expandable in Lidstone series. J. Math. Anal. Appl. 1974, 47, 626–632. [Google Scholar] [CrossRef]
  4. Boas, R.P. Representation of functions by Lidstone series. Duke Math. J. 1943, 10, 239–245. [Google Scholar]
  5. Boas, R.P.; Buck, R.C. Polynomial Expansions of Analytic Functions, 2nd ed.; Springer: Berlin, Germany, 1964. [Google Scholar]
  6. Golightly, G.O. Coefficients in sine series expansions of special entire functions. Huston J. Math. 1988, 14, 365–410. [Google Scholar]
  7. Leeming, D.; Sharma, A. A generalization of the class of completely convex functions. Symp. Inequalities 1972, 3, 177–199. [Google Scholar]
  8. Portisky, H. On certain polynomial and other approximations to analytic functions. Proc. Natl. Acad. Sci. USA 1930, 16, 83–85. [Google Scholar] [CrossRef]
  9. Schoenberg, I. On certain two-point expansions of integral functions of exponential type. Bull. Am. Math. Soc. 1936, 42, 284–288. [Google Scholar] [CrossRef]
  10. Widder, D. Completely convex functions and Lidstone series. Trans. Am. Math. Soc. 1942, 51, 387–398. [Google Scholar] [CrossRef]
  11. Ismail, M.; Mansour, Z.S. q-analogs of Lidstone expansion theorem, two point Taylor expansion theorem, and Bernoulli polynomials. Anal. Appl. 2018, 17, 1–47. [Google Scholar] [CrossRef]
  12. AL-Towailb, M. A generalization of the q-Lidstone series. AIMS Math. J. 2022, 7, 9339–9352. [Google Scholar] [CrossRef]
  13. AL-Towailb, M.; Mansour, Z.S. The q-Lidstone series involving q-Bernoulli and q-Euler polynomials generated by the third Jackson q-Bessel function. Khayyam J. Math. 2022; accepted. [Google Scholar]
  14. Mansour, Z.S.; AL-Towailb, M. The Complementary q-Lidstone Interpolating Polynomials and Applications. Math. Comput. Appl. 2020, 25, 34. [Google Scholar] [CrossRef]
  15. Al-Towailb, M. A q-Difference Equation and Fourier Series Expansions of q-Lidstone Polynomials. Symmetry 2022, 14, 782. [Google Scholar] [CrossRef]
  16. AL-Towailb, M.; Mansour, Z.S. Conditional expanding of functions by q-Lidstone series. Axiom 2023, 12, 22. [Google Scholar] [CrossRef]
  17. Mansour, Z.S.; AL-Towailb, M. q-Lidstone polynomials and existence results for q-boundary value problems. Bound Value Probl. 2017, 2017, 178. [Google Scholar] [CrossRef]
  18. Jackson, F.H. On q-functions and a certain difference operator. Trans. Roy. Soc. Edinb. 1908, 46, 64–72. [Google Scholar] [CrossRef]
  19. Ayman Mursaleen, M.; Serra-Capizzano, S. Statistical Convergence via q-Calculus and a Korovkin’s Type Approximation Theorem. Axioms 2022, 11, 70. [Google Scholar] [CrossRef]
  20. Kac, V.; Cheung, P. Quantum Calculus; Universitext; Springer: New York, NY, USA, 2002. [Google Scholar]
  21. Hadid, S.B.; Ibrahim, R.W.; Shaher, M. Multivalent functions and differential operator extended by the quantum calculus. Fractal Fract. 2022, 6, 354. [Google Scholar] [CrossRef]
  22. Ali, I.; Malghani, Y.A.K.; Hussain, S.M.; Khan, N.; Ro, J.-S. Generalization of k-Uniformly Starlike and Convex Functions Using q-Difference Operator. Fractal Fract. 2022, 6, 216. [Google Scholar] [CrossRef]
  23. Aldawish, I.; Ibrahim, R.W. Solvability of a new q-differential equation related to q-differential inequality of a special type of analytic functions. Fractal Fract. 2021, 5, 228. [Google Scholar] [CrossRef]
  24. Vivas-Cortez, M.; Aamir Ali, M.; Kashuri, A.; Bashir Sial, I.; Zhang, Z. Some New Newton’s Type Integral Inequalities for Co-Ordinated Convex Functions in Quantum Calculus. Symmetry 2020, 12, 1476. [Google Scholar] [CrossRef]
  25. Gasper, G.; Rahman, M. Basic Hypergeometric Series, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
  26. Jackson, F.H. On q-definite integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Al-Towailb, M.; Mansour, Z.S.I. A q-Analog of the Class of Completely Convex Functions and Lidstone Series. Axioms 2023, 12, 412. https://doi.org/10.3390/axioms12050412

AMA Style

Al-Towailb M, Mansour ZSI. A q-Analog of the Class of Completely Convex Functions and Lidstone Series. Axioms. 2023; 12(5):412. https://doi.org/10.3390/axioms12050412

Chicago/Turabian Style

Al-Towailb, Maryam, and Zeinab S. I. Mansour. 2023. "A q-Analog of the Class of Completely Convex Functions and Lidstone Series" Axioms 12, no. 5: 412. https://doi.org/10.3390/axioms12050412

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop